Almost every piano is slightly out of tune, on purpose.
Modern Western tuning splits the octave into 12 exactly equal steps, each a ratio of the twelfth root of two. The cost is that only the octave is acoustically pure; every other interval is fractionally off. The first person to calculate it precisely was a Ming dynasty prince, in 1584.
Musical intervals that sound sweetest correspond to simple ratios of frequency. The trouble is that these pure intervals don't fit together neatly: tune a keyboard so that some keys sound perfect and others will sound badly out of tune. Equal temperament solves this by compromise. It divides the octave into 12 identical semitones, each raising the frequency by a factor of about 1.05946, the twelfth root of two. Twelve steps multiply to exactly two, a perfect octave.
The price is spread evenly. The fifths and fourths of equal temperament are almost indistinguishable from pure ones, but the thirds and sixths are noticeably further off. In return, a musician can play in any key and move freely between them, and every key sounds the same.
The first precise calculation is usually credited to Zhu Zaiyu, a prince of the Ming court who spent some thirty years developing an idea of his father's. He published an exact numerical specification in a vast compendium on music and pitch in 1584, and built sets of bamboo pipes to demonstrate it; in 1890 a Brussels museum curator made pipes to his specifications and found them accurate. The Flemish scholar Simon Stevin reached a slightly less precise solution independently soon after. Whether Zhu's work reached Europe is unknown.
In practice, lute and guitar players had favoured equal tuning since the Renaissance. Its adoption elsewhere was contested, and some theorists, such as Giuseppe Tartini, objected that it spoiled the purity of chords. But its freedom to modulate proved irresistible, and it has dominated Western music since the 18th century. Other systems exist: Arabic theory uses 24 equal steps, and composers have written for 19 and 31. Equal temperament is a triumph of arithmetic over acoustics.
Source: Wikipedia — Equal temperament · Text summarised from Wikipedia (CC BY-SA 4.0)
